Optimal. Leaf size=139 \[ -\frac{2 (B d-A e)}{e (a+b x)^{3/2} \sqrt{d+e x} (b d-a e)}-\frac{4 \sqrt{d+e x} (a B e-4 A b e+3 b B d)}{3 \sqrt{a+b x} (b d-a e)^3}+\frac{2 \sqrt{d+e x} (a B e-4 A b e+3 b B d)}{3 e (a+b x)^{3/2} (b d-a e)^2} \]
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Rubi [A] time = 0.0866917, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {78, 45, 37} \[ -\frac{2 (B d-A e)}{e (a+b x)^{3/2} \sqrt{d+e x} (b d-a e)}-\frac{4 \sqrt{d+e x} (a B e-4 A b e+3 b B d)}{3 \sqrt{a+b x} (b d-a e)^3}+\frac{2 \sqrt{d+e x} (a B e-4 A b e+3 b B d)}{3 e (a+b x)^{3/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{A+B x}{(a+b x)^{5/2} (d+e x)^{3/2}} \, dx &=-\frac{2 (B d-A e)}{e (b d-a e) (a+b x)^{3/2} \sqrt{d+e x}}-\frac{(3 b B d-4 A b e+a B e) \int \frac{1}{(a+b x)^{5/2} \sqrt{d+e x}} \, dx}{e (b d-a e)}\\ &=-\frac{2 (B d-A e)}{e (b d-a e) (a+b x)^{3/2} \sqrt{d+e x}}+\frac{2 (3 b B d-4 A b e+a B e) \sqrt{d+e x}}{3 e (b d-a e)^2 (a+b x)^{3/2}}+\frac{(2 (3 b B d-4 A b e+a B e)) \int \frac{1}{(a+b x)^{3/2} \sqrt{d+e x}} \, dx}{3 (b d-a e)^2}\\ &=-\frac{2 (B d-A e)}{e (b d-a e) (a+b x)^{3/2} \sqrt{d+e x}}+\frac{2 (3 b B d-4 A b e+a B e) \sqrt{d+e x}}{3 e (b d-a e)^2 (a+b x)^{3/2}}-\frac{4 (3 b B d-4 A b e+a B e) \sqrt{d+e x}}{3 (b d-a e)^3 \sqrt{a+b x}}\\ \end{align*}
Mathematica [A] time = 0.0699987, size = 125, normalized size = 0.9 \[ -\frac{2 \left (A \left (-3 a^2 e^2-6 a b e (d+2 e x)+b^2 \left (d^2-4 d e x-8 e^2 x^2\right )\right )+B \left (3 a^2 e (2 d+e x)+2 a b \left (d^2+5 d e x+e^2 x^2\right )+3 b^2 d x (d+2 e x)\right )\right )}{3 (a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 177, normalized size = 1.3 \begin{align*} -{\frac{16\,A{b}^{2}{e}^{2}{x}^{2}-4\,Bab{e}^{2}{x}^{2}-12\,B{b}^{2}de{x}^{2}+24\,Aab{e}^{2}x+8\,A{b}^{2}dex-6\,B{a}^{2}{e}^{2}x-20\,Babdex-6\,B{b}^{2}{d}^{2}x+6\,A{a}^{2}{e}^{2}+12\,Aabde-2\,A{b}^{2}{d}^{2}-12\,B{a}^{2}de-4\,Bab{d}^{2}}{3\,{a}^{3}{e}^{3}-9\,{a}^{2}bd{e}^{2}+9\,a{b}^{2}{d}^{2}e-3\,{b}^{3}{d}^{3}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 14.1161, size = 682, normalized size = 4.91 \begin{align*} \frac{2 \,{\left (3 \, A a^{2} e^{2} -{\left (2 \, B a b + A b^{2}\right )} d^{2} - 6 \,{\left (B a^{2} - A a b\right )} d e - 2 \,{\left (3 \, B b^{2} d e +{\left (B a b - 4 \, A b^{2}\right )} e^{2}\right )} x^{2} -{\left (3 \, B b^{2} d^{2} + 2 \,{\left (5 \, B a b - 2 \, A b^{2}\right )} d e + 3 \,{\left (B a^{2} - 4 \, A a b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{3 \,{\left (a^{2} b^{3} d^{4} - 3 \, a^{3} b^{2} d^{3} e + 3 \, a^{4} b d^{2} e^{2} - a^{5} d e^{3} +{\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{3} +{\left (b^{5} d^{4} - a b^{4} d^{3} e - 3 \, a^{2} b^{3} d^{2} e^{2} + 5 \, a^{3} b^{2} d e^{3} - 2 \, a^{4} b e^{4}\right )} x^{2} +{\left (2 \, a b^{4} d^{4} - 5 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} + a^{4} b d e^{3} - a^{5} e^{4}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.82576, size = 763, normalized size = 5.49 \begin{align*} -\frac{2 \,{\left (B b^{2} d e - A b^{2} e^{2}\right )} \sqrt{b x + a}}{{\left (b^{3} d^{3}{\left | b \right |} - 3 \, a b^{2} d^{2}{\left | b \right |} e + 3 \, a^{2} b d{\left | b \right |} e^{2} - a^{3}{\left | b \right |} e^{3}\right )} \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} - \frac{4 \,{\left (3 \, B b^{\frac{13}{2}} d^{3} e^{\frac{1}{2}} - 4 \, B a b^{\frac{11}{2}} d^{2} e^{\frac{3}{2}} - 5 \, A b^{\frac{13}{2}} d^{2} e^{\frac{3}{2}} - 6 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2} B b^{\frac{9}{2}} d^{2} e^{\frac{1}{2}} - B a^{2} b^{\frac{9}{2}} d e^{\frac{5}{2}} + 10 \, A a b^{\frac{11}{2}} d e^{\frac{5}{2}} + 12 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2} A b^{\frac{9}{2}} d e^{\frac{3}{2}} + 3 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{4} B b^{\frac{5}{2}} d e^{\frac{1}{2}} + 2 \, B a^{3} b^{\frac{7}{2}} e^{\frac{7}{2}} - 5 \, A a^{2} b^{\frac{9}{2}} e^{\frac{7}{2}} + 6 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2} B a^{2} b^{\frac{5}{2}} e^{\frac{5}{2}} - 12 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2} A a b^{\frac{7}{2}} e^{\frac{5}{2}} - 3 \,{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{4} A b^{\frac{5}{2}} e^{\frac{3}{2}}\right )}}{3 \,{\left (b^{2} d^{2}{\left | b \right |} - 2 \, a b d{\left | b \right |} e + a^{2}{\left | b \right |} e^{2}\right )}{\left (b^{2} d - a b e -{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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